# Solidity Mathematical Functions

In this article, we will explore some of the most common Solidity mathematical functions and how they can be utilized to create robust and reliable smart contracts.

Solidity has various mathematical functions that allow you to perform complex mathematical operations within smart contracts.

These functions are essential to ensure the accuracy and security of transactions executed on the blockchain.

## Basic Arithmetic Operators

Using Solidity Arithmetic Operators , you can perform simple mathematical operations such as addition, subtraction, multiplication, and division.

#### Example:

pragma solidity ^0.8.0;contract SolidityTest{ function add_Values(uint mrx,uint ample)public pure returns (uint){ return mrx+ample;}function find_Difference(uint mrx,uint ample) public pure returns (uint){ return mrx-ample;}function find_Product(uint mrx,uint ample) public pure returns (uint){ return mrx*ample;}function calculate_division(uint mrx,uint ample) public pure returns (uint){ return mrx/ample;} }

## Built-In Mathematical Functions

Solidity has a range of mathematical functions that can be used to perform operations such as

These functions are similar to those found in other programming languages like Python and Java.

The following are the built in function solidity provides us:

 Functions Overview The addmod() function takes three inputs, mrx, ample, and mrx_Ample, then adds mrx and ample, and then takes the modulus relative to mrx_Ample as (mrx+ample)% mrx_Ample. mulmod() This function takes three inputs mrx,ample, and mrx_Ample. It multiplies mrx and ample, then takes the modulus of mrx_Ample using the formula (mrx*ample)% mrx_Ample. min() The min() function returns the minimum value among two unsigned integers. max() The max() function returns the maximum value among two unsigned integers. Sqrt() returns the square root of a user given unsigned integer. log2() The log2() function returns the log2 value of any unsigned integer taken as input. Average() The Average() function is used to add two user given integers and then divide the corresponding result with 2. pow() It displays the power of a base up to the limit assigned by the user. exp() The exp() function computes 2 raised to the power of any unsigned integer taken as input.

The given example shows the use of all the built-in functions in solidity:

#### Example:

pragma solidity ^0.8.0;import "hardhat/console.sol"; import { Math } from "@openzeppelin/contracts/utils/math/Math.sol";contract solidityTest { function find_addmod(uint mrx, uint ample, uint k) public pure returns (uint) { return addmod(mrx, ample, k); } function find_mulmod(uint mrx, uint ample, uint k) public pure returns (uint) { return mulmod(mrx,ample, k); } function find_min(uint mrx, uint ample) external pure returns (uint) { return Math.min(mrx, ample); } function find_max(uint mrx, uint ample) public pure returns (uint) { return Math.max(mrx, ample); } function find_sqrt(uint mrx) public pure returns (uint) { return Math.sqrt(mrx); }function find_logarithm2(uint mrx) public pure returns(uint){ return Math.log2(mrx); }function find_Average(uint mrx,uint ample) public pure returns(uint){return Math.average(mrx,ample); }function find_Power(uint mrx,uint ample) public pure returns(uint){ return Math.pow(mrx,ample); } function find_Exp(uint mrx) public pure returns(uint){return Math.exp(mrx); }}
Note: It is advisable to add functions in the Math library if it lacks any. Else you can replace your math library by the following code:

## Mr Examples Math Library Code

// OpenZeppelin Contracts (last updated v4.8.0) (utils/math/Math.sol)

pragma solidity ^0.8.0;

/**
* @dev Standard math utilities missing in the Solidity language.
*/
library Math {
enum Rounding {
Down, // Toward negative infinity
Up, // Toward infinity
Zero // Toward zero
}

/**
* @dev Returns the largest of two numbers.
*/
function max(uint256 a, uint256 b) internal pure returns (uint256) {
return a > b ? a : b;
}

/**
* @dev Returns the smallest of two numbers.
*/
function min(uint256 a, uint256 b) internal pure returns (uint256) {
return a < b ? a : b;
}

/**
* @dev Returns the average of two numbers. The result is rounded towards
* zero.
*/
function average(uint256 a, uint256 b) internal pure returns (uint256) {
// (a + b) / 2 can overflow.
return (a & b) + (a ^ b) / 2;
}

/**
* @dev Returns the ceiling of the division of two numbers.
*
* This differs from standard division with `/` in that it rounds up instead
* of rounding down.
*/
function ceilDiv(uint256 a, uint256 b) internal pure returns (uint256) {
// (a + b - 1) / b can overflow on addition, so we distribute.
return a == 0 ? 0 : (a - 1) / b + 1;
}

/**
* @notice Calculates floor(x * y / denominator) with full precision. Throws if result overflows a uint256 or denominator == 0
* @dev Original credit to Remco Bloemen under MIT license (https://xn--2-umb.com/21/muldiv)
* with further edits by Uniswap Labs also under MIT license.
*/
function mulDiv(
uint256 x,
uint256 y,
uint256 denominator
) internal pure returns (uint256 result) {
unchecked {
// 512-bit multiply [prod1 prod0] = x * y. Compute the product mod 2^256 and mod 2^256 - 1, then use
// use the Chinese Remainder Theorem to reconstruct the 512 bit result. The result is stored in two 256
// variables such that product = prod1 * 2^256 + prod0.
uint256 prod0; // Least significant 256 bits of the product
uint256 prod1; // Most significant 256 bits of the product
assembly {
let mm := mulmod(x, y, not(0))
prod0 := mul(x, y)
prod1 := sub(sub(mm, prod0), lt(mm, prod0))
}

// Handle non-overflow cases, 256 by 256 division.
if (prod1 == 0) {
return prod0 / denominator;
}

// Make sure the result is less than 2^256. Also prevents denominator == 0.
require(denominator > prod1);

///////////////////////////////////////////////
// 512 by 256 division.
///////////////////////////////////////////////

// Make division exact by subtracting the remainder from [prod1 prod0].
uint256 remainder;
assembly {
// Compute remainder using mulmod.
remainder := mulmod(x, y, denominator)

// Subtract 256 bit number from 512 bit number.
prod1 := sub(prod1, gt(remainder, prod0))
prod0 := sub(prod0, remainder)
}

// Factor powers of two out of denominator and compute the largest power of two divisors of denominator. Always >= 1.
// See https://cs.stackexchange.com/q/138556/92363.

// Does not overflow because the denominator cannot be zero at this stage in the function.
uint256 twos = denominator & (~denominator + 1);
assembly {
// Divide denominator by twos.
denominator := div(denominator, twos)

// Divide [prod1 prod0] by twos.
prod0 := div(prod0, twos)

// Flip twos such that it is 2^256 / twos. If two is zero, then it becomes one.
twos := add(div(sub(0, twos), twos), 1)
}

// Shift in bits from prod1 into prod0.
prod0 |= prod1 * twos;

// Invert denominator mod 2^256. Now that denominator is an odd number, it has an inverse modulo 2^256 such
// that denominator * inv = 1 mod 2^256. Compute the inverse by starting with a seed that is correct for
// four bits. That is, denominator * inv = 1 mod 2^4.
uint256 inverse = (3 * denominator) ^ 2;

// Use the Newton-Raphson iteration to improve the precision. Thanks to Hensel's lifting lemma, this also works
// in modular arithmetic, doubling the correct bits in each step.
inverse *= 2 - denominator * inverse; // inverse mod 2^8
inverse *= 2 - denominator * inverse; // inverse mod 2^16
inverse *= 2 - denominator * inverse; // inverse mod 2^32
inverse *= 2 - denominator * inverse; // inverse mod 2^64
inverse *= 2 - denominator * inverse; // inverse mod 2^128
inverse *= 2 - denominator * inverse; // inverse mod 2^256

// Because the division is now exact we can divide by multiplying with the modular inverse of the denominator.
// This will give us the correct result modulo 2^256. Since the preconditions guarantee that the outcome is
// less than 2^256, this is the final result. We don't need to compute the high bits of the result and prod1
// is no longer required.
result = prod0 * inverse;
return result;
}
}

/**
* @notice Calculates x * y / denominator with full precision, following the selected rounding direction.
*/
function mulDiv(
uint256 x,
uint256 y,
uint256 denominator,
Rounding rounding
) internal pure returns (uint256) {
uint256 result = mulDiv(x, y, denominator);
if (rounding == Rounding.Up && mulmod(x, y, denominator) > 0) {
result += 1;
}
return result;
}

/**
* @dev Returns the square root of a number. If the number is not a perfect square, the value is rounded down.
*
* Inspired by Henry S. Warren, Jr.'s "Hacker's Delight" (Chapter 11).
*/
function sqrt(uint256 a) internal pure returns (uint256) {
if (a == 0) {
return 0;
}

// For our first guess, we get the biggest power of 2 which is smaller than the square root of the target.
//
// We know that the "msb" (most significant bit) of our target number `a` is a power of 2 such that we have
// `msb(a) <= a < 2*msb(a)`. This value can be written `msb(a)=2**k` with `k=log2(a)`.
//
// This can be rewritten `2**log2(a) <= a < 2**(log2(a) + 1)`
// → `sqrt(2**k) <= sqrt(a) < sqrt(2**(k+1))`
// → `2**(k/2) <= sqrt(a) < 2**((k+1)/2) <= 2**(k/2 + 1)`
//
// Consequently, `2**(log2(a) / 2)` is a good first approximation of `sqrt(a)` with at least 1 correct bit.
uint256 result = 1 << (log2(a) >> 1);

// At this point `result` is an estimation with one bit of precision. We know the true value is a uint128,
// since it is the square root of a uint256. Newton's method converges quadratically (precision doubles at
// every iteration). We thus need at most 7 iteration to turn our partial result with one bit of precision
// into the expected uint128 result.
unchecked {
result = (result + a / result) >> 1;
result = (result + a / result) >> 1;
result = (result + a / result) >> 1;
result = (result + a / result) >> 1;
result = (result + a / result) >> 1;
result = (result + a / result) >> 1;
result = (result + a / result) >> 1;
return min(result, a / result);
}
}

/**
* @notice Calculates sqrt(a), following the selected rounding direction.
*/
function sqrt(uint256 a, Rounding rounding) internal pure returns (uint256) {
unchecked {
uint256 result = sqrt(a);
return result + (rounding == Rounding.Up && result * result < a ? 1 : 0);
}
}

function pow(uint256 base, uint256 exponent) internal pure returns (uint256) {
if (exponent == 0) {
return 1;
} else if (exponent % 2 == 0) {
uint256 temp = pow(base, exponent / 2);
return temp * temp;
} else {
uint256 temp = pow(base, (exponent - 1) / 2);
return base * temp * temp;
}
}

function exp(uint256 x) internal pure returns (uint256) {
uint256 result = 1;
uint256 factorial = 1;

for (uint256 i = 1; i <= 10; i++) {
factorial *= i;
result += (x ** i) / factorial;
}

return result;
}

/**
* @dev Return the log in base 2, rounded down, of a positive value.
* Returns 0 if given 0.
*/
function log2(uint256 value) internal pure returns (uint256) {
uint256 result = 0;
unchecked {
if (value >> 128 > 0) {
value >>= 128;
result += 128;
}
if (value >> 64 > 0) {
value >>= 64;
result += 64;
}
if (value >> 32 > 0) {
value >>= 32;
result += 32;
}
if (value >> 16 > 0) {
value >>= 16;
result += 16;
}
if (value >> 8 > 0) {
value >>= 8;
result += 8;
}
if (value >> 4 > 0) {
value >>= 4;
result += 4;
}
if (value >> 2 > 0) {
value >>= 2;
result += 2;
}
if (value >> 1 > 0) {
result += 1;
}
}
return result;
}

/**
* @dev Return the log in base 2, following the selected rounding direction, of a positive value.
* Returns 0 if given 0.
*/
function log2(uint256 value, Rounding rounding) internal pure returns (uint256) {
unchecked {
uint256 result = log2(value);
return result + (rounding == Rounding.Up && 1 << result < value ? 1 : 0);
}
}

/**
* @dev Return the log in base 10, rounded down, of a positive value.
* Returns 0 if given 0.
*/
function log10(uint256 value) internal pure returns (uint256) {
uint256 result = 0;
unchecked {
if (value >= 10**64) {
value /= 10**64;
result += 64;
}
if (value >= 10**32) {
value /= 10**32;
result += 32;
}
if (value >= 10**16) {
value /= 10**16;
result += 16;
}
if (value >= 10**8) {
value /= 10**8;
result += 8;
}
if (value >= 10**4) {
value /= 10**4;
result += 4;
}
if (value >= 10**2) {
value /= 10**2;
result += 2;
}
if (value >= 10**1) {
result += 1;
}
}
return result;
}

/**
* @dev Return the log in base 10, following the selected rounding direction, of a positive value.
* Returns 0 if given 0.
*/
function log10(uint256 value, Rounding rounding) internal pure returns (uint256) {
unchecked {
uint256 result = log10(value);
return result + (rounding == Rounding.Up && 10**result < value ? 1 : 0);
}
}

/**
* @dev Return the log in base 256, rounded down, of a positive value.
* Returns 0 if given 0.
*
* Adding one to the result gives the number of pairs of hex symbols needed to represent `value` as a hex string.
*/
function log256(uint256 value) internal pure returns (uint256) {
uint256 result = 0;
unchecked {
if (value >> 128 > 0) {
value >>= 128;
result += 16;
}
if (value >> 64 > 0) {
value >>= 64;
result += 8;
}
if (value >> 32 > 0) {
value >>= 32;
result += 4;
}
if (value >> 16 > 0) {
value >>= 16;
result += 2;
}
if (value >> 8 > 0) {
result += 1;
}
}
return result;
}

/**
* @dev Return the log in base 10, following the selected rounding direction, of a positive value.
* Returns 0 if given 0.
*/
function log256(uint256 value, Rounding rounding) internal pure returns (uint256) {
unchecked {
uint256 result = log256(value);
return result + (rounding == Rounding.Up && 1 << (result * 8) < value ? 1 : 0);
}
}
}

### Conclusion

Solidity provides built-in math functions that are essential for performing arithmetic and other mathematical operations within smart contracts.

Besides simple arithmetic operations like addition and multiplication, these functions also include exponential and logarithmic functions.

The correct use of these functions is essential to prevent errors and ensure the correct operation of smart contracts. In spite of their computational expense, these functions are essential to many applications and should be used as efficiently as possible.

Using Solidity’s built-in math functions, you can create smart contracts that are capable of handling a variety of transactions in various industries, including finance, gaming, and supply chain management.

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